I don't think it's exactly what you might be after, but in Firefox there is an add-on called "Scrapbook" that I use to manually save copies of web pages I bring up on my screen. They are saved in HTML format in an indexed, searchable library.

Also, don't forget your history file and that your browser cache might be automatically saving a lot of stuff, so a cache reader (Nirsoft has some good ones) can come in very handy.

I suppose web spiders would do this, though I don't use any so can't go into specifics. Also, you asked for something that automatically saves all your activity - I haven't seen a web spider that does this, they need to be assigned a site/pages to leech.